Algebraic approach to a $d$-dimensional matrix Hamiltonian with so($d+1)$ symmetry

A novel spin-extended so($d+1$,1) algebra is introduced and shown to provide an interesting framework for discussing the properties of a $d$-dimensional matrix Hamiltonian with spin 1/2 and so($d+1$) symmetry. With some $d+2$ additional operators, spanning a basis of an so($d+1$,1) irreducible representation, the so($d+1$,1) generators provide a very easy way for deriving the integrals of motion of the matrix Hamiltonian in Sturm representation. Such integrals of motion are then transformed into those of the matrix Hamiltonian in Schr\"odinger representation, including a Laplace-Runge-Lenz vector with spin. This leads to a derivation of the latter, as well as its properties in a more extended algebraic framework.